Math 204, Spring 2016
Honors
Mathematics II
Instructor
John E. McCarthy
Class
MTuThF
11.00-12.00 in 207 Cupples
I on Monday, in 199 Cupples I on Tuesday,Thursday, Friday
Discussion Section F 2.00-3.00 in Room 199, Cupples I
JM Office
105
Cupples I
JM Office Hours
M 12.00-1.00, Tu
10.00-11.00, Th
12.00-1.00, and by appointment
Phone
935-6753
Teaching
Assistant
Tokio
Sasaki
TS Office Hours M 3.00-4.00, W 12.00-1.00, Th 3.00-4.00, F 3.00-4.00 in Room 6, Cupples I
Exams There will be three exams in the course:
1) Exam 1 In class.
Friday February 19.
2) Exam 2
Take Home. Due
Monday April 2.
3) Exam 3
Final exam. Tuesday May 10,
10.30-12.30.
Homework
There will be weekly homework sets during the semester,
assigned on
Tuesday and due the following Tuesday.
Homework
1 due January 26.
Homework 2
due February 2.
Homework 3
due February 9.
Homework 4
due February 16.
Homework 5
due February 23.
Homework 6
due March 1.
Homework 7
due March 8.
Homework 8
due March 22.
Homework 9
due March 29.
Homework 10
due April 5.
Homework 11
due April 12.
Homework 12
due April 19.
Homework 13
due April 26.
Homework 14.
Prerequisites
Math 203, or permission of instructor.
Description
This is the second half of a one-year calculus sequence for
first year
students with a strong interest in mathematics.
The course will be challenging, with an emphasis on rigor and proofs.
If you complete the year-long sequence, you will not only cover all of
Math
233, but most of Math
318 and Math 310, and some
of Math 309.
More importantly, you will have learned what real mathematics is!
In the first semester, we covered the basics of proofs
(including
addressing why proofs are important, and not just a formal exercise),
revisited one variable calculus, and spent some time on vectors.
In
the second semester, we will cover matrices, functions of several
variables, partial and total derivatives, multiple integrals,
line and surface integrals, Green's, Stokes's and Gauss's theorems.
Content
Here is a very
tentative schedule. We will not stick
to it closely.
Week 1: Linear transformations and matrices. Determinants.
Week 2: Eigenvalues and eigenvectors.
Week
3: Solving linear systems. Rank
nullity theorem.
Week 4:
Limits and continuity in R^n. Partial and directional derivatives.
Week 5: Total Derivatives. Chain rule.
Week 6:
Higher order partial derivatives. Extremum problems in several variables.
Week 7: Hessians. Lagrange multipliers.
Week 8: Line integrals.
Week 9: Multiple Integrals.
Week 10: Green's Theorem
Week 11: Change of variables in multiple integrals. Spherical and cylindrical coordinates.
Week 12: Surface integrals
Week 13:
Stokes and Gauss's theorems.
Week 14:
Preceding material will take more
than 13 weeks to cover; I am not sure exactly where it will expand,
Basis for Grading
Each midterm and the homework will be 20% of your grade,
the final will be 40%. If you do well on the final, this grade can be
substituted for one of your midterms.
Homework
Homework is an extremely important part of the course. Whilst
talking to
other people about it is not dis-allowed,
too often
this degenerates into one person solving the problem, and other people
copying
them (often justified to themselves by saying "I provide the ideas, X
does
the details" - but the details are the key. If you can't translate the
idea into a real proof, you don't understand the material well enough).
So I
shall introduce the following rules:
(a) You can only talk to
some-one else about a problem
if you have made a genuine effort to solve it
yourself.
(b) You must write up the solutions on your own. Suspiciously similar
write-ups
will receive 0 points.
Class
I do expect you to come to class every day, and to participate in class discussions. I also expect you to stay abreast of the material we are covering, and may call on you at any time to answer a question.
The Discussion Sections are very strongly recommended, but I
understand some
students will have unavoidable conflicts with them.
Class etiquette: don't be disruptive or discourteous. No beeping, ringing, crunching, rustling, leaving early or arriving late. No texting, sleeping, checking your phone.
Texts
Transition to Higher
Mathematics: Structure and Proof
by Bob Dumas and John McCarthy ( Available
free here)
Calculus,
Volume I by Tom Apostol,
(Wiley) Second Edition, 1967
Calculus,
Volume II by Tom Apostol,
(Wiley) Second Edition, 1969
Multivariable Mathematics
by
Theodore Shifrin (Wiley 2005)
Note on the texts: The two books by Apostol
are
very expensive. They are not required texts for the course, though they
will be
useful.
I recommend buying used copies if you can.
There will be a copy of both Volume I and II on two-hour reserve at the
Library.
The book by Shifrin (also not required, and also, unfortunately, expensive) is less dense than Apostol.
Additional Reading
Vector Calculus by J. Marsden and A. Tromba